Biassociahedra Revisited Joint work with Samson Saneblidze Ron - PowerPoint PPT Presentation

Left-iterated S-U Diagonal � Define ∆ ( 0 ) : = Id and P � ∆ P ⊗ Id ⊗ n − 1 � ∆ ( n ) ∆ ( n − 1 ) : = P P � Denote the up-rooted n -leaf corolla by � n � Think of ∆ ( n − 1 ) ( � m + 1 ) as a subcomplex of ( P m ) × n P � � vertices of ∆ ( n − 1 ) ( � m + 1 ) is a subposet of ( S m ) × n � X n m = P

Left-iterated S-U Diagonal � Define ∆ ( 0 ) : = Id and P � ∆ P ⊗ Id ⊗ n − 1 � ∆ ( n ) ∆ ( n − 1 ) : = P P � Denote the up-rooted n -leaf corolla by � n � Think of ∆ ( n − 1 ) ( � m + 1 ) as a subcomplex of ( P m ) × n P � � vertices of ∆ ( n − 1 ) ( � m + 1 ) is a subposet of ( S m ) × n � X n m = P � x ∈ X n m ↔ n × 1 matrix of up-rooted trees with m + 1 leaves

Vertices as Matrices �� � �� � ∆ ( 1 ) P ( � ) = � ⊗ � implies X 2 1 = �

Vertices as Matrices �� � �� � ∆ ( 1 ) P ( � ) = � ⊗ � implies X 2 1 = � � � � ∆ ( 1 ) = ⊗ + ⊗ implies P � � X 2 2 = , ,

Vertices as Matrices �� � �� � ∆ ( 1 ) P ( � ) = � ⊗ � implies X 2 1 = � � � � ∆ ( 1 ) = ⊗ + ⊗ implies P � � X 2 2 = , , � Vertex of particular interest is 3 ⊂ ( P 3 ) x 4 ∈ X 4

Vertices as Matrices � � � n + 1 �� vertices of ∆ ( m − 1 ) is a subposet of ( S n ) × m � Y m n = P

Vertices as Matrices � � � n + 1 �� vertices of ∆ ( m − 1 ) is a subposet of ( S n ) × m � Y m n = P � y ∈ Y m n ↔ 1 × m matrix of down-rooted trees with n + 1 leaves

Vertices as Matrices � � � n + 1 �� vertices of ∆ ( m − 1 ) is a subposet of ( S n ) × m � Y m n = P � y ∈ Y m n ↔ 1 × m matrix of down-rooted trees with n + 1 leaves � ∆ ( 2 ) P ( � ) = � ⊗ � ⊗ � implies Y 3 1 = { [ � � � ] }

Vertices as Matrices � � � n + 1 �� vertices of ∆ ( m − 1 ) is a subposet of ( S n ) × m � Y m n = P � y ∈ Y m n ↔ 1 × m matrix of down-rooted trees with n + 1 leaves � ∆ ( 2 ) P ( � ) = � ⊗ � ⊗ � implies Y 3 1 = { [ � � � ] } � � = � ∆ ( 1 ) ⊗ + ⊗ implies P � � Y 2 2 = , ,

Vertices as Matrices � � � n + 1 �� vertices of ∆ ( m − 1 ) is a subposet of ( S n ) × m � Y m n = P � y ∈ Y m n ↔ 1 × m matrix of down-rooted trees with n + 1 leaves � ∆ ( 2 ) P ( � ) = � ⊗ � ⊗ � implies Y 3 1 = { [ � � � ] } � � = � ∆ ( 1 ) ⊗ + ⊗ implies P � � Y 2 2 = , , � Vertex of particular interest is ∈ Y 4 3

Vertices as Matrix Products � Trees with levels factor uniquely as matrix products of levels

Vertices as Matrix Products � Trees with levels factor uniquely as matrix products of levels � x ∈ X n m factors uniquely as a matrix product x = x 1 · · · x m ∈ X 2 2

Vertices as Matrix Products � Trees with levels factor uniquely as matrix products of levels � x ∈ X n m factors uniquely as a matrix product x = x 1 · · · x m ∈ X 2 2 � y ∈ Y m factors uniquely as a matrix product y = y n · · · y 1 n ∈ Y 2 2

Transverse Product � A bisequence matrix of graphs α y i x j has the form   α y 1 α y 1 · · · x 1 x p   . . . .   . . α y q α y q · · · x 1 x p

Transverse Product � A bisequence matrix of graphs α y i x j has the form   α y 1 α y 1 · · · x 1 x p   . . . .   . . α y q α y q · · · x 1 x p � A transverse product of bisequence matrices has form   α y 1 p � � p · · · α y q : = α y 1   . β q β q p . · · ·   . x 1 x p β q x 1 · · · β q α y q x p p

Block Transverse Product � A typical Block Transverse Product (BTP) has the form α 1 α 1 2 1 β 3 β 3 β 3 α 5 α 5 1 2 3 2 1 β 1 β 1 β 1 α 4 α 4 1 2 3 2 1 α 3 α 3 2 1       α 1 α 1  � �  � � 2 1   β 3 β 3  β 3  α 5 α 5   2 1 3 1 2   α 4 α 4 =   2 1   � � � � � � � � β 1 β 1 β 1 α 3 α 3 2 1 3 1 2

Block Transverse Product � A typical Block Transverse Product (BTP) has the form α 1 α 1 2 1 β 3 β 3 β 3 α 5 α 5 1 2 3 2 1 β 1 β 1 β 1 α 4 α 4 1 2 3 2 1 α 3 α 3 2 1       α 1 α 1  � �  � � 2 1   β 3 β 3  β 3  α 5 α 5   2 1 3 1 2   α 4 α 4 =   2 1   � � � � � � � � β 1 β 1 β 1 α 3 α 3 2 1 3 1 2 � The BTP acts associatively on bisequence matrices

Vertices as BTPs � � � � � � � � � � � � [ � ] [ � ] [ �� ] � = = � � � �

Vertices as BTPs � � � � � � � � � � � � [ � ] [ � ] [ �� ] � = = � � � � � Vertices of KK n , m “generated by” X n m − 1 × Y m n − 1 ⊂ P m + n − 2

Vertices as BTPs � � � � � � � � � � � � [ � ] [ � ] [ �� ] � = = � � � � � Vertices of KK n , m “generated by” X n m − 1 × Y m n − 1 ⊂ P m + n − 2 �� � � � [ �� ] � X 2 1 × Y 2 1 = generates the vertices of KK 2 , 2 �

Vertices as BTPs � � � � � � � � � � � � [ � ] [ � ] [ �� ] � = = � � � � � Vertices of KK n , m “generated by” X n m − 1 × Y m n − 1 ⊂ P m + n − 2 �� � � � [ �� ] � X 2 1 × Y 2 1 = generates the vertices of KK 2 , 2 � �� � �� � � � �� � � � � � [ � � � ] , [ � � � ] , � X 2 2 × Y 3 1 = � � � � � � � � �� � � � � [ � � � ] generates the vertices of KK 2 , 3 � � �

Matrix Transposition � A = [ a ij ] is { � , � } -matrix; rows contain � exactly once

Matrix Transposition � A = [ a ij ] is { � , � } -matrix; rows contain � exactly once � B = [ b ij ] is { � , � } -matrix; columns contain � exactly once

Matrix Transposition � A = [ a ij ] is { � , � } -matrix; rows contain � exactly once � B = [ b ij ] is { � , � } -matrix; columns contain � exactly once � ( A , B ) is an ( i , j ) -edge pair if � a ij � � b ij b i , j + 1 = � a i + 1 , j

Matrix Transposition � A = [ a ij ] is { � , � } -matrix; rows contain � exactly once � B = [ b ij ] is { � , � } -matrix; columns contain � exactly once � ( A , B ) is an ( i , j ) -edge pair if � a ij � � b ij b i , j + 1 = � a i + 1 , j    � � � � � � � �    ,  � Typical ( 1 , 1 ) -edge pair: � � � � � �

Matrix Transposition � A = [ a ij ] is { � , � } -matrix; rows contain � exactly once � B = [ b ij ] is { � , � } -matrix; columns contain � exactly once � ( A , B ) is an ( i , j ) -edge pair if � a ij � � b ij b i , j + 1 = � a i + 1 , j    � � � � � � � �    ,  � Typical ( 1 , 1 ) -edge pair: � � � � � �    �  � � � � � � �    ,  � Not an edge pair: � � � � � �

Matrix Transposition � If ( A , B ) is an ( i , j ) -edge pair...

Matrix Transposition � If ( A , B ) is an ( i , j ) -edge pair... � A i ∗ is the matrix obtained from A by deleting the i th row

Matrix Transposition � If ( A , B ) is an ( i , j ) -edge pair... � A i ∗ is the matrix obtained from A by deleting the i th row � B ∗ j is the matrix obtained from B by deleting the j th column

Matrix Transposition � If ( A , B ) is an ( i , j ) -edge pair... � A i ∗ is the matrix obtained from A by deleting the i th row � B ∗ j is the matrix obtained from B by deleting the j th column � The ( i , j ) - transposition of AB is B ∗ j A i ∗

Matrix Transposition � If ( A , B ) is an ( i , j ) -edge pair... � A i ∗ is the matrix obtained from A by deleting the i th row � B ∗ j is the matrix obtained from B by deleting the j th column � The ( i , j ) - transposition of AB is B ∗ j A i ∗ � � � ( 1 , 1 ) -transpose [ �� ] [ � ] [ � ] � Ex: ⇒ �

Matrix Transposition � If ( A , B ) is an ( i , j ) -edge pair... � A i ∗ is the matrix obtained from A by deleting the i th row � B ∗ j is the matrix obtained from B by deleting the j th column � The ( i , j ) - transposition of AB is B ∗ j A i ∗ � � � ( 1 , 1 ) -transpose [ �� ] [ � ] [ � ] � Ex: ⇒ � � If c = C 1 · · · C r and ( C k , C k + 1 ) is an ( i , j ) -edge pair, define ij ( c ) : = C 1 · · · C ∗ j k + 1 C i ∗ T k k · · · C r

Matrix Transposition � Iterate T on X n m − 1 × Y m n − 1 in all possible ways: � � � � T k t i t j t · · · T k 1 � u ∈ X n m − 1 × Y m Z n , m = i 1 j 1 ( u ) n − 1

Matrix Transposition � Iterate T on X n m − 1 × Y m n − 1 in all possible ways: � � � � T k t i t j t · · · T k 1 � u ∈ X n m − 1 × Y m Z n , m = i 1 j 1 ( u ) n − 1 � The vertex poset of P n + m − 2 extends to X n m − 1 × Y m n − 1 � Z n , m

Matrix Transposition � Iterate T on X n m − 1 × Y m n − 1 in all possible ways: � � � � T k t i t j t · · · T k 1 � u ∈ X n m − 1 × Y m Z n , m = i 1 j 1 ( u ) n − 1 � The vertex poset of P n + m − 2 extends to X n m − 1 × Y m n − 1 � Z n , m � Vertices of KK 2 , 3 :

Matrix Transposition � Iterate T on X n m − 1 × Y m n − 1 in all possible ways: � � � � T k t i t j t · · · T k 1 � u ∈ X n m − 1 × Y m Z n , m = i 1 j 1 ( u ) n − 1 � The vertex poset of P n + m − 2 extends to X n m − 1 × Y m n − 1 � Z n , m � Vertices of KK 2 , 3 :

Matrix Transposition � Iterate T on X n m − 1 × Y m n − 1 in all possible ways: � � � � T k t i t j t · · · T k 1 � u ∈ X n m − 1 × Y m Z n , m = i 1 j 1 ( u ) n − 1 � The vertex poset of P n + m − 2 extends to X n m − 1 × Y m n − 1 � Z n , m � Vertices of KK 2 , 3 :

A Vertex to be Discarded � Our 2011 construction admits all elements of Z n , m

A Vertex to be Discarded � Our 2011 construction admits all elements of Z n , m � If m + n ≥ 8 , certain elements of Z n , m must be discarded

A Vertex to be Discarded � Our 2011 construction admits all elements of Z n , m � If m + n ≥ 8 , certain elements of Z n , m must be discarded � Consider the following vertex u ∈ X 4 3 × Y 4 3 :       � � � � � �   � � � �� � � � � � � � � � � �       � � � � [ � � �� ]         � � � � � �       � � � � � � � � � � � � � � =

A Vertex to be Discarded � T 3 1 , 1 ( u ) =     � � �     � � � �� �� � � � � �  �   � �  � � � � � � [ � � �� ]         � � �     � � � � � � � � � � � � � =

A Vertex to be Discarded � z = T 2 1 , 1 T 3 1 , 1 ( u ) =   �       � � � �� � � � � � � � �  �     � �   � �   � � �  [ � � �� ] �   � � � � � � � � � � � � = ∈ Z 4 , 4

A Vertex to be Discarded � z = T 2 1 , 1 T 3 1 , 1 ( u ) =   �       � � � �� � � � � � � � �  �     � �   � �   � � �  [ � � �� ] �   � � � � � � � � � � � � = ∈ Z 4 , 4 � The 2 nd matrix above is not a bisequence matrix, however...

A Vertex to be Discarded � ∃ ! balanced factorization with bisequence indecomposable factors   � �     [ � � �� ] ABC =  �  �

A Vertex to be Discarded � ∃ ! balanced factorization with bisequence indecomposable factors   � �     [ � � �� ] ABC =  �  � � A vertex is admissible if balanced factorization is ∆ P - coherent

A Vertex to be Discarded � ∃ ! balanced factorization with bisequence indecomposable factors   � �     [ � � �� ] ABC =  �  � � A vertex is admissible if balanced factorization is ∆ P - coherent � We now realize that the product AB fails to be ∆ P -coherent

A Vertex to be Discarded � Write the entries of AB in their balanced factorizations:   �  �    AB = = �   �

A Vertex to be Discarded � Write the entries of AB in their balanced factorizations:   �  �    AB = = �   � � Up-rooted trees with same row (input) leaf sequences are � � and

A Vertex to be Discarded � Write the entries of AB in their balanced factorizations:   �  �    AB = = �   � � Up-rooted trees with same row (input) leaf sequences are � � and is not an edge of ∆ P ( 12 | 3 ) ⊂ ( P 2 × P 1 ) × 2 � ⊗

A Vertex to be Discarded � Write the entries of AB in their balanced factorizations:   �  �    AB = = �   � � Up-rooted trees with same row (input) leaf sequences are � � and is not an edge of ∆ P ( 12 | 3 ) ⊂ ( P 2 × P 1 ) × 2 � ⊗ � Since AB fails to be ∆ P -coherent, vertex ABC is discarded